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The separation concept is essential in convex programming. In particular, the convex sets separation. In this work the Hahn-Banach theorem is presented, with great generality, together with an important separation theorem, its consequence. Then are followed the necessary mathematical steps towards the minimax theorem, an important result in the applications of Game Theory to economics, finance and management.Keywords: Hahn-Banach theorem, separation theorems, convex programming, minimax theorem